I have a query on a Random process derived from Markov process. I have stuck in this problem for more than 2 weeks.(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex]r(t)[/itex] be a finite-state Markov jump process described by

\begin{alignat*}{1}

\lim_{dt\rightarrow 0}\frac{Pr\{r(t+dt)=j/r(t)=i\}}{dt} & =q_{ij}

\end{alignat*}

when [itex]i \ne j[/itex], and where [itex]q_{ij}[/itex] is the transition rate and represents the probability per time unit that [itex]r(t)[/itex] makes a transition from state $i$ to a

state $j$. Now, let [itex]r(\rho(t))[/itex] be a random process derived from [itex]r(t)[/itex] depending on a parameter [itex]\rho(t)[/itex], which is defined by

\begin{alignat*}{1}

\frac{d}{dt}\rho(t)=f(r(\rho(t))),\qquad\rho(0)=0

\end{alignat*}

Here [itex]f(.)[/itex] is a piecewise continuous function depending on [itex]r(\rho(t))[/itex]

with range space as [itex]\mathbb{R}[/itex], a set of Real numbers. In this case can we describe the random process [itex]r(\rho(t))[/itex] as

\begin{alignat*}{1}

\lim_{dt\rightarrow 0}\frac{\mathrm{Pr}\{r(\rho(t+dt))=j/r(\rho(t))=i\}}{\rho(t+dt)-\rho(t)} =q_{ij},\qquad i\ne j\\

\end{alignat*}

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# Random process derived from Markov process

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